Finite Group Databasebeta

← #22566022

Subgroups — #22566022

Order 8576.

508 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524022C211yesmaximal, characteristic
2#16863335C212yesmaximal, complemented
3#16863329C214yesmaximal, complemented
4#16863316C212yesmaximal, complemented
5#16863269C212yesmaximal
6#16863259C212yesmaximal
7#16863256C211yesmaximal, characteristic
8C22×C4⋊C4×C67C211yesmaximal, characteristic
9C22×Dic134C2211yescharacteristic
10#14524022—24no—
11C2×C4⋊Dic67C2211yescharacteristic
12C2×C4⋊Dic67C2212yes—
13C2×Dic67⋊C4C2212yes—
14C2×Dic67⋊C4C2214yes—
15C2×C22⋊Dic67C2212yes—
16C2×C22⋊Dic67C2212yes—
17C67⋊#95C2212yes—
18C67⋊#95C2214yes—
19C67⋊#95C2214yes—
20C67⋊#95C2214yes—
21C23×C4×C67C2211yescharacteristic
22C23×C4×C67C2211yescharacteristic
23C23×C4×C67C2211yescharacteristic
24C2×C4⋊C4×C67—24no—
25C2×C4⋊C4×C67C2212yes—
26C2×C4⋊C4×C67C2212yes—
27C2×Dic134—22no—
28C2×Dic134—42no—
29C22×Dic67C2312yes—
30C22×Dic67C2314yes—
31C22×Dic67C2312yes—
32C4⋊Dic67—22no—
33C4⋊Dic67—42no—
34Dic67⋊C4—24no—
35Dic67⋊C4—44no—
36C22⋊Dic67—24no—
37C22⋊Dic67—24no—
38C24×C67C2311yescharacteristic
39C22×C4×C67—22no—
40C22×C4×C67—22no—
41C22×C4×C67—22no—
42C22×C4×C67—42no—
43C22×C4×C67—42no—
44C22×C4×C67D414yes—
45C22×C4×C67Q814yescomplemented
46C22×C4×C67C2312yes—
47C22×C4×C67C2311yescharacteristic
48C22×C4×C67C2311yescharacteristic
49C22×C4×C67C2312yes—
50C4⋊C4×C67—24no—
51C4⋊C4×C67—24no—
52C23×C67—22no—
53C23×C67—22no—
54C23×C67C2×D412yes—
55C23×C67C2×Q812yes—
56C23×C67#4112yes—
57C23×C67C2411yescharacteristic
58C2×C4×C67—28no—
59C2×C4×C67—22no—
60C2×C4×C67—22no—
61C2×C4×C67—22no—
62C2×C4×C67—22no—
63C2×C4×C67—22no—
64C2×C4×C67—24no—
65C2×C4×C67—44no—
66C2×C4×C67—44no—
67C2×C4×C67C2×D412yes—
68C2×C4×C67C2×D412yes—
69C2×C4×C67C2×Q814yes—
70C2×Dic67—22no—
71C2×Dic67—22no—
72C2×Dic67—24no—
73C2×Dic67—24no—
74C2×Dic67—24no—
75C2×Dic67—24no—
76C2×Dic67—24no—
77Dic134—44no—
78C22×C67—22no—
79C22×C67—22no—
80C22×C67—24no—
81C22×C67—24no—
82C22×C67C22⋊Q814yes—
83C22×C67C22×D411yescharacteristic
84C22×C67C22×Q811yescharacteristic
85C22×C67C2×#4111yescharacteristic
86C22×C67H(5,𝔽2)12yes—
87C22×C672−1+412yes—
88C4×C67—24no—
89C4×C67—22no—
90C4×C67—22no—
91C4×C67—42no—
92Dic67—42no—
93Dic67—42no—
94Dic67—44no—
95C2×C67—24no—
96C2×C67C2×C22⋊Q811yescharacteristic
97C2×C67#53411yescharacteristic
98C2×C67#53611yescharacteristic
99C2×C67#53912yes—
100C2×C67#54412yes—
101#2627—671nomaximal, Sylow
102C67#262711yescharacteristic, complemented, Sylow
103C2×C22⋊Q8—671no—
104C22×C4⋊C4D6711yescharacteristic
105#400—672no—
106#398—674no—
107#395—672no—
108#385—672no—
109#381—672no—
110#380—671no—
111C22×Q8—671no—
112C23×C4C2×D6711yescharacteristic
113C23×C4C2×D6711yescharacteristic
114C23×C4C2×D6711yescharacteristic
115C22⋊Q8—1344no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—674no—
118C2×C4⋊C4—672no—
119C2×C4⋊C4—672no—
120C2×C4⋊C4—671no—
121C2×C4⋊C4C2×D6712yes—
122C2×C4⋊C4C2×D6712yes—
123C2×C22⋊C4—672no—
124C2×C22⋊C4—672no—
125#95—672no—
126#95—674no—
127#95—674no—
128#95—674no—
129C24C22×D6711yescharacteristic
130C2×Q8—2682no—
131C2×Q8—1342no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—672no—
138C22×C4—674no—
139C22×C4—672no—
140C22×C4Dic13414yescomplemented
141C22×C4C22⋊D6714yes—
142C22×C4C22×D6712yes—
143C22×C4C22×D6711yescharacteristic
144C22×C4C22×D6711yescharacteristic
145C22×C4C22×D6712yes—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—2684no—
149C4⋊C4—2682no—
150C4⋊C4—1344no—
151C4⋊C4—1342no—
152C22⋊C4—1344no—
153C22⋊C4—1344no—
154C23—22no—
155C23—22no—
156C23C2×C22⋊D6712yes—
157C23C2×Dic13412yes—
158C23#1176162812yes—
159C23C23×D6711yescharacteristic
160Q8—2684no—
161C2×C4—28no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—1342no—
171C2×C4—1342no—
172C2×C4—1344no—
173C2×C4—1344no—
174C2×C4—1344no—
175C2×C4—1344no—
176C2×C4—1344no—
177C2×C4C2×C22⋊D6712yes—
178C2×C4C2×C22⋊D6712yes—
179C2×C4C2×Dic13414yes—
180C22—24no—
181C22—22no—
182C22—22no—
183C22—24no—
184C22#1452402214yes—
185C22C22×C22⋊D6711yescharacteristic
186C22C22×Dic13411yescharacteristic
187C22C2×#1176162811yescharacteristic
188C22#1452408812yes—
189C22#1452409112yes—
190C4—24no—
191C4—22no—
192C4—22no—
193C4—42no—
194C4—2682no—
195C4—2682no—
196C4—2684no—
197C2—24no—
198C2C2×#1452402211yescharacteristic
199C2#1686395511yescharacteristic
200C2#1686397011yescharacteristic
201C2#1686399312yes—
202C2#1686402712yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC22×C4⋊C4×C67C2
hypercenterC22×C4⋊C4D67
hypocenterC67#2627
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #16863335C2semidirect
3. #16863329C2semidirect
4. #16863316C2semidirect
45. C22×C4×C67Q8semidirect
102. C67#2627semidirect
140. C22×C4Dic134semidirect